A circle's center is at #(3 ,5 )# and it passes through #(2 ,8 )#. What is the length of an arc covering #(5pi ) /4 # radians on the circle?

Question

A circle's center is at #(3 ,5   )# and it passes through #(2    ,8   )#.  What is the length of an arc covering #(5pi ) /4  # radians on the circle?

Evelyn Arboleda · Accepted Answer

#(5sqrt(10)pi)/4# Part 1 If a circle has a center at #(3,5)# and passes through #(2,8)# it has a radius of #r=sqrt((3-2)^2+(5-8)^2)= sqrt(1+9)=sqrt(10)# Part 2 An arc with an angle of #k# radians has a length of #k/(2pi) xx "circumference of the circle"# but since the #"circumference of the circle" = 2pir# the length of an arc with an angle of #k# radians is k * r# Part 3 For the given circle and arc the arc length is #(5pi)/4xxsqrt(10)#

Parker Allen · Answer

undefined The radius of the circle is the distance between its center and any point on the circle. Using the distance formula, the radius is √((3-2)² + (5-8)²) = √(1² + (-3)²) = √(1 + 9) = √10.

The circumference of a circle is given by the formula C = 2πr, where r is the radius. So, the circumference of this circle is 2π√10.

The length of an arc on a circle is given by the formula L = rθ, where r is the radius and θ is the angle in radians subtended by the arc at the center of the circle. Therefore, the length of the arc covering (5π/4) radians on this circle is √10 * (5π/4) = 5√10π/4.